Bifurcations and dynamical behaviors for a generalized delayed-diffusive Maginu model

This paper is committed to study the dynamical behaviors of a generalized Maginu model with discrete time delay. We investigate the stability of the positive equilibrium and the existence of periodic solutions bifurcating from the positive equilibrium. Further, by using the center manifold theorem and the normal form theory, we derive the precise condition to judge the bifurcation direction and the stability of the bifurcating periodic solutions. Also, we deduce the exact condition to determine the Turing instability of the Hopf bifurcating periodic solutions for diffusive system. Numerical simulations are used to support our theoretical analysis.